sbrml part5 introduction to bipedal walking · 2 sensor based robotic manipulation and locomotion...

1 Sensor Based Robotic Manipulation and Locomotion

Introduction to Bipedal Walking

Dr.-Ing. Christian OttDLR - Institute for Robotics and Mechatronics

For full lecture on humanoid systems see Christian‘s lecture at EIModeling and Control of Humanoid Walking Robots

http://www.robotic.dlr.de/chr

1

DLR 02/05/2012


Motivation

Honda Asimo

Bipedalism:• Stepping on different heights and over obstacles• Small region of support compared to wheeled robots• Humanoid body structure allows to act

in human environments• Smallest number of legs required for

standing, walking, running• Fundamental research:

Control, planning, mech. designUnderstanding (human) balance

• „Technological competition“: Sony, Honda, Toyota, Samsung,Aldebaran, Boston Dynamics, …

• …


Humanoid Balance

Vestibular system

Vision

Somatosensory System

Proprioceptivesensors

IMU

Vision

force sensors

joint sensing

“Balance” is a generic term describing the ability to control the body posturein order to prevent falling.


Humanoid Balance

Small push:Ankle strategy

force controlZMP control

(Zero Moment Point)

angular momentum control

Medium push:Hip strategy

Large Push: Step strategy

Human

Robot

Strategies for human push recovery:


Balancing Walking

Types of bipedal gaitStatic walking: Ground projection of center of motion (COM) never leaves support polygon

Dynamic walking:Def A: Ground projection leaves support polygon during motion„Def B: Walking with underactuation“ (e.g. point foot walking)

Running: includes flight phase


Models

Multi-Body-Models Conceptual Models

Fixed Base Models(predefined contact state)

Floating Base Model Walking Running

Dynamical Models (Mechanical)

Complexity

Specialization


Floating base model

)3(SEHb Qq

n

111 SSST n

Configuration Space: )3(SEQ

Using local coordinates: n6

6bx


Free-Floating vs. Fixed Base Models

Fixed base modelsIn each contact state the model is different:

• Single support (right, left)• Double support• Heel Off• Toe Touch Down• …

Transition between contact states

double supportparallel kinematicsover-constrained

need for passive joints

single supportserial kin. chainor tree structure

Free-floating model

Components:• Lagrangian dynamics• Constraints due to contact forces• Transition equations (impacts)

underactuated

Planning & control must ensure that the considered contact state is valid! ground reaction force must fulfill constraints for balancing


Zero Moment Point[Vukobratovic and Stepanenko,1972]

)(x1x 2x

F

ZMP as a ground reference point: Distributed ground reaction force under the supporting foot can be replaced by a single force F acting at the ZMP, called ground reaction force.

z

x

),( yx

y

ZMP = CoP (Center of Pressure)

p1x 2x

F0

ZMP

CoP


Definition of the Zero Moment Point (ZMP)

Planar single support:

Distributed floor reaction force under the supporting foot can be replaced by a single force acting at the ZMP.

[Vukobratovic and Stepanenko,1972]

2

1

2

1

)(

)(0)( x

x

x

x

dxx

dxxxpp

z

x

)(x1x 2x

210)( xpxx

F

2

1

)()()(x

xdxxpxp

ZMP = CoP (Center of Pressure)

Extension to two dimensions is straight forward


Some facts about the ZMP• Can ZMP leave the support polygon? NO• Can ZMP location be used as a stability criterion NO

• If ZMP reaches the border of the support polygon foot rotation possible.

• ZMP is defined on flat contact (no uneven surface).• ZMP gives no information about sliding.

)(x1x 2x

F

Measurement e.g. by Force/Torque Sensor

),,(0)()(!

ssssss fppfppp

z

sfs

spp

How to obtain ZMP in practice?


First usage of the ZMP• Motion of the legs is predefined.• Upper body controls the ZMP in the center of the supporting foot

ensure proper foot contact during walking


Conceptual Models for Walking

Cart table model Inverted pendulum model

Ground reaction force should stay within the stance area

Ground reaction force stays at the hinge point of the pendulum

Can be derived from the general model:

• Approximation of angular momentum

• Limited motion (no vertical COM motion)

Basis for many successful walking robots

NAO HRP-2 ASIMO


Mass Concentrated Model for Linear Inverted Pendulum

Forces in the linear inverted pendulum (LIP) model

p

xc

zc

Mg

xcM

F

pccgc xz

x

pf p p

ppfp

c


mass concentrated model

Forces in the LIP model

p

xc

zc

Mg

xcM

F

Effect of an additional hip torque

p

xcM

F

pccgc xz

x

z

xz

x Mcpc

cgc

Mg


mass concentrated model

Strategies for gait stabilization: Effect of an additional hip torque

p

Mg

xcM

F

z

xz

x Mcpc

cgc

1. Controlling ZMP (constraints!)

2. Angular momentum control

3. Step adaptation


Mass concentrated model for ZMP Control

p

c

gcccp xz

xx

xcInverted Pendulum Model [Sugihara]

xxz

x pccgc

p

c

xx pc xx cp

Cart Table Model [Kajita]

Simplifying assumptions• robot mass concentrated in the center of mass (CoM)• CoM height cz is kept constant

We have a simple relation between the motion of the CoM and the ZMP


ZMP based walking pattern: basic scheme

Footstep planning

Walking Pattern

Generator

CoM Inverse

kinematics

Joint Position Control

dp dc dq

Image copied without explicit permission from Workshop material “Overview of ZMP-based Biped Walking” at Dynamic Walking conference 2008, by S. Kajita.

Simple solution: Use hip motion (+offset) instead of CoM

Here, CoM is controlled


DLR Robot Control Based on Conceptual Models

Footstep Generation

Pattern Generation

x

pZMP-COMStabilizer

dxPos. Controlled

Robot

e.g. LQR Preview Control [Kajita, 2003]

Model Predictive Control [Wieber, 2006]

realtime

F

Automatica 2010

ZMP is controlled


Torque Based BalancingA Unified Approach for

Grasping and Balancing


Torque based balancingAssumptions:

Joint Torque Control

COM and hip orientation can be measured in a world-fixed frame (via inertia measurement unit – IMU - measurement)

extFMg





Control Approach:

1. Compute desired force on the COM (according to compliant behavior)

COMF

extFMg





Control Approach:

1. Compute desired force on the COM(according to compliant behavior)

2. Distribute COM force to contact points COMF

extFMg




COM and hip orientation can be measured in a world-fixed frame (via IMU measurement)

Control Approach:

1. Compute desired force on the COM(according to compliant behavior)

2. Distribute COM force to contact points

3. Realize contact forces via joint torques

COMF

extFMg


Force distribution: grasping and balancing are very similar problems!

oFo

f1 f2

W

f1 f2

Fo

Grasping and Balancing


Force Distribution in Grasping

F

FGGFGFGWO

1

111Net wrench acting on the object:

Grasp Map

if

)3(seFC

Well studied problem in grasping: Find contact wrenches such that a desired net wrench on the object is achieved.

FCFC

)3(se

friction cone


Balancing & Posture Control

Compliant COM control [Hyon & Cheng, 2006]

Compliant trunk orientation Control =>

)()( dDdPCOM ccKccKMgF

Mg

extF

COMF

HIPT

)3(SOR

HIPT extT

),( HIPCOMd TFW Desired wrench:

Quaternion based orientation compliance control(see passivity based Cartesian impedance control)


Force distribution

HIPT

COMF

f

fGGWd

1

1

ii

ii Rp

RG

ˆ

3if

Relation between balancing wrench & contact forces

Constraints:• Unilateral contact: (implicit handling of ZMP constraints)• Friction cone constraints

0, zif

Cf

T

F

GG


Force distribution

HIPT

COMF

f

fGGWd

1

1

ii

ii Rp

RG

ˆ

3if

Relation between balancing wrench & contact forces

Constraints:• Unilateral contact: (implicit handling of ZMP constraints)• Friction cone constraints

0, zif

Formulation as a constraint optimization problem

Cf

23

22

21minarg CCTHIPCFCOMC ffGTfGFf

T

F

GG

321


ForceDistribution

Torque based balancing

Force Mapping

TorqueControl

RobotDynamics

Object ForceGeneration

IMU

cf

q

for orientation control and COM computation


ForceDistribution

Torque based balancing

Force Mapping

TorqueControl

RobotDynamics

Object ForceGeneration

IMU

cf

q

for orientation control and COM computation

COM

Contact forces


Summary

1. Consistent treatment of COM and posture control (useful for manipulation, bipedal vehicles)

2. Implicit handling of ZMP via constraints in the force optimization

3. Utilizes a formulation from grasping theory: Allows for generalization to multi-contact situations

4. Controller is independent of precise knowledge about foot contact(however, IMU data is important!)

Outlook:- Extension to multi-contact interactions- Extension to walking


Overview

1. Fundamentals about bipedal walking

2. Time based walking control ZMP based control

3. Limit cycle based walking Passive dynamic walking


Passive Dynamic Walking„Passive Dynamic Walking“

Dynamics Control

• Careful mechanic design:knee retraction, foot shape, trunk, elastic elements

• Analysis: Limit cycle (Poincare Map), impacts


Passive Dynamic Walking

„Dynamic Walking“

Actuation + Dynamics

„Passive Dynamic Walking“

Dynamics Control


Conceptual Models: RunningTemplates

• Template for control

• Template for design

[Geyer, Seyfarth, Blikhan, 2006]

Role of compliance for human walking/running.

[Oscar Pistorius][A. Sato, Mc Gill Univ.]


Legged Hopping Robots

[Raibert, MIT]

Three part control:

1. control of hopping height (during stance)

2. control forward velocity via foot positioning

3. control of body orientation by servoing the hip


[Raibert, MIT]


That's all Folks!

sbrml part5 introduction to bipedal walking · 2 sensor based robotic manipulation and locomotion...

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